In some fanfictions, his family might be involved in dark magic or have a bad reputation. So, Harry Potter refuses them to uphold his own moral values. He is known for his strong sense of right and wrong, and if his family goes against that, he would distance himself.
Another reason could be that the family in the fanfiction is trying to control his life. Harry has always been an independent spirit. If they are overly protective or manipulative, he would likely refuse them. For example, they might try to stop him from going on adventures or making his own decisions about his magical education.
One possible reason could be the way the Dursleys treated him. They were cruel, made him live in a cupboard, and constantly belittled him. This kind of mistreatment over the years could easily lead to hatred. Another aspect could be the lack of connection to his wizarding family that he was not aware of for a long time. He might feel angry at his parents for leaving him in such a situation without any real protection from the Dursleys.
One possible reason could be that the author of the fanfiction is portraying him in a way that goes against his character. Harry is known for his integrity and if the fanfic makes him do unethical things, he might seem to'refuse to talk' in that narrative.
In some fanfictions, Harry might leave his family because of a prophecy or a special mission. He might feel that he has to go it alone in order to fulfill this prophecy or complete the mission successfully. His family might not understand or be able to support him in this, so he decides to leave. Also, there could be jealousy or resentment within the family that makes the environment unbearable for Harry.
One possible reason could be the overwhelming burden of his destiny. Constantly facing dark forces and the pressure of saving the wizarding world might have worn him down. Another reason could be the loss of his loved ones. For example, if in the fanfiction Hermione or Ron were killed, that could shatter his spirit.
In some fanfiction, Harry might renounce his name to explore a new identity. He has been through so much as Harry Potter, and by giving up his name, he can break free from the predefined roles and destinies associated with it. He could then embark on a journey of self - discovery, free from the constraints of the name that has both defined and haunted him.
One possible reason could be that in the fanfiction, Harry might have discovered some dark secrets about his parents that made him feel betrayed. For example, if he found out they had some hidden and unethical magical experiments in their past.
One possible reason could be that he might feel abandoned. Growing up without them, he could resent the fact that they left him an orphan. He might think that if they were really so great, they would have found a way to survive and be there for him.
Well, perhaps Harry Potter as a character represents hope and kindness. Allowing bullying fanfic would be like tainting his image. In the original story, Harry has been a victim of bullying himself, so it would be inconsistent for him to be associated with content that promotes bullying. Also, the Harry Potter franchise is aimed at a wide audience, including children, and bullying fanfic doesn't fit with the family - friendly and positive nature of the overall franchise.
Perhaps he discovers some secrets about his parents that he doesn't like. For example, he might find out that they were involved in some dangerous or unethical magic before. This new knowledge could make him angry at them for putting themselves and their family in jeopardy.
Maybe he was tired of the constant danger and responsibility that came with magic. In the original series, he faced so many threats. In fanfiction, he might just want a normal life away from all that.